Modeling of geometric errors of linear guideway and their influence on joint kinematic error in machine tools

Precision Engineering 36 (2012) 369–378

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Review

Modeling of geometric errors of linear guideway and their influence on joint kinematic error in machine tools Paweł Majda ∗ West Pomeranian University of Technology, Faculty of Mechanical Engineering and Mechatronics, Al. Piastów 19, 70-310 Szczecin, Poland

a r t i c l e

i n f o

Article history: Received 8 July 2011 Received in revised form 8 November 2011 Accepted 4 February 2012 Available online 15 February 2012 Keywords: Machine tool accuracy Guideway geometric error FEM

a b s t r a c t This paper presents the problems of the geometric accuracy of machine tools. The analytical and experimental examinations were carried out for a table in which guideway geometric errors may result in significant deformations. The main aim was to propose a method of analytical examination of the influence of geometric errors in linear guideway on joint kinematic errors. The proposed method served to isolate and simulate geometric errors, one of the causes of volumetric errors in machine tools. This approach helped to understand and interpret the results of experimental examinations of angular kinematic errors (pitch, yaw, roll) obtained for a real machine tool. The results helped to verify the hypothesis that the deformation of a table may be a significant source of errors in volumetric error models. One of the final conclusions indicated that off-line compensation of some characteristics of angular kinematic errors in machine tools may be unjustified. © 2012 Elsevier Inc. All rights reserved.

Contents 1. 2. 3. 4. 5. 6. 7.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model for linear guideway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The results of calculations for static characteristics of a linear guideway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model for guideway geometric error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation of joint kinematic errors of a machine tool table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental research of angular kinematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Guideway geometric errors are one of the causes of undesirable differences between the nominal and real position and orientation of a tool in relation to the workpiece. These differences in the entire workspace of a machine are known as volumetric error. Beside the geometric errors the influences on volumetric errors have: temperature changes which cause thermal deformations, deformations of machine tool parts caused by operating loads, the properties of the axis controls. It may be (in example for precision machine tool with hydrostatic linear guides [1]) that geometric errors have a small share in the total volumetric error. Then a question arises that is there an explanation and a point

∗ Tel.: +48 91 449 42 66; fax: +48 91 449 44 42. E-mail address: [email protected] 0141-6359/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2012.02.001

369 370 371 372 372 376 377 377 377

of geometric errors compensation in such cases? In addition, the problem can be dealt with the distinction of machine tool working states, (i) with the cutting process taken into account, and (ii) without cutting process. In the first case, compensation may be unprofitable, because due to the presence of other sources of errors in the machine it is not really possible to obtain the higher dimensional and shape accuracy of the workpieces. The second case we have to deal with when measuring the geometrical specification on the machine tool using a 3D measurement touch probe. Such probes use machine tool displacement measurement systems as a length standard. It is one of the flaws of this solution, because the measurement accuracy is determined by the accuracy of the machine tool itself. In such case the geometric errors participation is greater in the total balance of volumetric error. Then the geometric error compensation can give the desired increase of the accuracy of measurements made directly on the machine tool.

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The knowledge of volumetric error is used in software offline compensation of machine tool errors, thus increasing their positioning accuracy. The efficiency of such compensation is especially effective for large machine tools [2]. Large machine tool tables are bigger then 1200 mm × 600 mm, medium sized machine tools tables are smaller then 1200 mm × 600 mm [3]. One of the characteristic properties of large machine tools is their kinematic structure, usually of a portal/gantry type, where a tool path runs in many axes against a fixed workpiece. In this case, regardless of the measurement method and determination of volumetric error, the fixed table on which the workpiece lies is also a measurement datum for the measured machine tool errors. Volumetric error is determined in relation to that datum. A similar procedure is applied for medium-sized machine tools. The fact that mediumsized machine tools usually have a movable table as a measurement datum should also be taken into account. Using direct measurement [4] of machine tool errors, e.g. by laser inferometer or electronic level, the elements of the measuring system are thus located on a movable table. In indirect measurements [4] the situation is similar, for example in the method proposed by Yang et al. [5] in which a ball bar test is used, with one end situated in different positions on the table. The methodology of measurements using a ball plate artefact, used by Bringmann and Knapp [6], and other material artefacts (Woody et al. [2]; Choi et al. [7]), also requires the location of measuring system elements on the table. Effective tools for mapping of volumetric errors using a lasertracer system (developed under the supervision of Schwenke [8,9]) also require the location of the measuring equipment on the table. A question arises, (i) whether this situation may generate errors in the determination of volumetric error, and (ii) whether methods verified for large-sized machine tools can be adopted for medium-sized machine tools. The range of analytical and experimental studies presented in the further part of this paper should help answer to these questions. Apart from the methodology of measurements, the accuracy of software compensation of machine tools errors is very much affected by the manner and accuracy of volumetric error modeling. The principal assumption of modeling is the ability to determine volumetric error based on the characteristics of joint kinematic errors, i.e. translation errors (positional, vertical and horizontal straightness) and angular errors (pitch, yaw, roll) associated with the kinematics of the studied solid, most often treated as a rigid body [10–12]. It is desirable that the characteristics of kinematic errors have the characteristics of systematic errors. The model assumptions are then fulfilled and volumetric error compensation is justified. One of the most interesting papers on the verification of modeling assumptions is by Ekinci and Mayer [13]. The authors performed a detailed analysis of a basic guideway. They examined the relationships between straightness and angular kinematic errors for different ratios of carriage length to the wavelength of a guideway geometric error. For a two-dimensional model, they analytically proved and then experimentally verified the relationship between angular errors and respective straightness errors. They showed how significant errors result from strict conversion of angular errors into straightness errors. The conclusions of the authors [13] were the result of a detailed analysis at an elementary level, i.e. on the level of elementary causes of volumetric errors. The causes were obviously guideway geometric errors. It must be emphasized here that many authors had previously uncritically used a simplified assumption that angular errors may be converted directly into straightness errors [14–16] or vice versa [17]. It is still an open question as to what extent such simplifications affect the accuracy of volumetric error determination. The direct inspiration for this paper were the observations made during modeling of volumetric error for a specific structure of a medium-sized milling machine (vertical machining centres) with a cross table. It was noticed that some characteristics of angular

kinematic errors measured on the milling machine table strongly depended on the location of measuring instruments. This effect should be unnoticeable if the assumption of ideal stiffness of the examined solids is to be permissible (correct) in a volumetric error model. The discussion presented in this paper continues the examination of problems from the paper [13], i.e. associated with the identification of sources of machine tool errors and the accuracy of volumetric errors models. Our simulation studies were extended for a three-dimensional case. The realization of such a task required the development of a modeling method which should enable an analytical examination of the effect of guideway geometric errors on joint kinematic errors, and thus on volumetric errors in machine tools. Based on literature, we developed a model of a single roller linear guideway. The manner of modeling is presented in Section 2. In order to verify the reliability of results obtained with the developed model, Section 3 presents the comparison of calculated characteristics of force–displacement and the characteristics provided by the manufacturer of the guideways. In the next stage in Section 4 we present the manner of modeling entire guideways allowing for guide rail geometric errors. Section 5 presents an example of a milling machine with a linear guideway including guide rail geometric errors. Based on the conclusions drawn after the performed simulation, an experimental examination was prepared, the results of which are presented and commented in Section 6. In this paper I assumed the definition of machine tools according to the division proposed by [13]. 2. Model for linear guideway In modeling, the finite elements method (FEM) [18] was used. The sub-assemblies of a guideway were discretized with solid elements, which made it possible to allow for a elasticity within the linear elastic range for carriage and guide rails. Considering the usefulness of the performed analytical examinations, it was deemed that omitting the elasticity of single rolling elements is a too great simplification, which decreases the reliability of the performed analyses. Below is the description of the applied models. A contact element was used to model the interactions between contact phenomena in the area of the roller patch – rolling element – roller patch, and the elasticity of a single rolling element. With regard to physical modeling, a rolling element is replaced by a spring element of unilateral action (compression only), which connects/joins a carriage with a guide rail at the contact points between rolling element and roller patch. It has a linear dimension – a length equal to the diameter of a rolling element. The idea of modeling is presented in Fig. 1 [19–21]. Considering geometric non-linearity (only compression) and physical non-linearity, the contact deformation of the single ball within two grooves can be found by Hertzian analysis as being dependent on the normal load F [19,21,22].



ı(F) =

g − 2 · ı

for :

F ≤0

compression

→∞

for :

F>0

tension

(1)

 where ı(F) = 1.41 ·  ·

3

F2 ·

2·D−d d·D



· 2·

1−2 E

2 ı(F), deformation

of a single ball; g, gap or preload (Fig. 2); F, force acting on a single ball; d, diameter of the ball; D, diameter of the groove; E, Young’s modulus of ball and groove material; , Poisson ratio of ball and groove material; and , parameter that depends on the ratio d and D. If D → ∞ (i.e. ball in contact with flat surface), then  → 1 Fig. 2 presents the effect of g parameter from Eq. (1) on the characteristics of contact deformation of a rolling element when g is lower than zero, gap occurs. When g is greater than zero, preload

P. Majda / Precision Engineering 36 (2012) 369–378

371

Fig. 1. Cross-section of the linear guideway – diagrams of models for solid structures and rolling elements.

of the contact element occurs. Parameter g was used in the model for guideway preload and for mapping/reproducing guide rail geometric errors. 3. The results of calculations for static characteristics of a linear guideway The object of the model and calculations was a single guideway, a carriage-guide rails type. The modeling was carried out in accordance with the description in Section 2. The parameters of Eq. (1) for contact elements which model rolling elements are: d = 3 mm, D = 1.01 d, E = 210 GPa,  = 0.3,  = 1.412. The analysis concerned the structure of a roller sub-assembly, size 25 (width of the rail at its base) and number 1651, according to the manufacturer’s product list [23]. Necessary geometric dimensions and preload for the carriage were also taken from the product list. Preload was 8% C; C is the dynamic capacity and equals 22.8 kN. Dividing the preload by the number of rolling elements gives force per rolling element. Based on the force and characteristic described by Eq. (1), parameter g is calculated. The parameter was used directly in the FEM guideway model, thus allowing for carriage preload. In the calculations the base of the guide rail was fixed and the concentrated force was applied in the model using a solid with ideal stiffness. The comparison of the calculations with the experimental characteristics provided by the guideway manufacturer [23] for various loads (upwards, downwards, and lateral) is shown in Fig. 3. This comparison of the aforementioned results with the FEM calculations shows a substantial conformity with data provided by the manufacturer. This applies especially to schemas of downward and lateral load. A somewhat worse consistency (greater calculation stiffness), especially for a force greater than 9 kN, was obtained for an upward load. The main source of this discrepancy

Fig. 3. The comparison of calculations for static guideway characteristics with characteristics provided by the producer [23] for various types of loads: F: (a) upward, (b) downward, and (c) lateral.

is probably the lack of models for screw joints. It should be noted that the characteristics were obtained on the basis of a computational model for a system with clearly defined assumptions and without the need to identify any of the parameters of the model. Therefore it must be considered that the achieved qualitative and quantitative conformity was high and proves the high reliability of the calculations using the aforementioned guideway model. An important advantage of this presented model is the possibility

Fig. 2. The effect of g parameter on the characteristics displacement–force of rolling element.

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Fig. 4. Example diagram of a model for straightness error in the system roller patch – rolling elements – roller patch, using contact elements.

of its use in spatial analysis including geometric errors of the guideways themselves.

4. Model for guideway geometric error One of the main purposes of this is was to propose a method of analysis of the effect of guideway geometric error on the characteristics of joint kinematic errors in machine tools. Theoretically, it is possible to create an FEM model which takes into account geometric errors in the form of an appropriate distribution of nodes belonging to the finite elements. However, since the geometry of shape errors (including the location of relevant nodes) in relation to the dimensions of the model is subject to marginally small changes, this approach would be questionable due to errors of discretization of the model [18], thus making the obtained results also questionable. Therefore, in this work, the proposed model has a constant topology of element nodes, and the mapping of geometric errors is performed by the application of values variable for the parameters of contact elements that model the rolling elements. Modeling guideways according to the manner presented in Section 2 provides such opportunities through the use of different values of g in Eq. (1). The idea of mapping the straightness error by giving different values of parameter g in the contact elements is shown in Fig. 4. The balance of internal forces in such a system will be reached after projection of the defined shape error. The modeling of geometric errors occurring in the linear guides components is based on differing the value of g parameter according to the assumed geometric error describing function. Taking into account the geometry of the rolling assembly the formulas for calculating of the g parameter are as follows:for g > 0 that is for the preload (extension)

 c 2 + (b + i ) − d

 2

gV =

b2 + (c + i ) − d

The adopted object of the model was a guideway with four carriages and a table – Fig. 6. Models for carriages and guide rails were made according to the technique described in Section 2. The configuration of table dimensions and guideway correspond to the machine tool visible in the photographs presented in the graphs in Figs. 10–12. A cast iron table was modeled in linear elastic – FEM. The model geometry was simplified. The thickness of the table was the result of optimization, which minimized the difference in k1 stiffness of a detailed model and k2 stiffness of a simplified model – Fig. 6a and b. The calculations allowed for the force of gravity. The base of the guide rails was fixed. The table had the ability to move freely along the rails, therefore in order to solve the system of equations in the FEM model, the movement had to be reduced through a model for a guide screw. A guide screw was modeled as a spring element, schematically shown in Fig. 6c. The stiffness of that element corresponds to the bar stiffness respective to the screw core diameter – 32 mm. In models for guideway geometric errors, according to the concept presented in Section 4, there is no need to model the entire guide rails, and there is no need to create multiple computational models for the different positions of the table against the guides. The representation of geometric errors is carried out through the use of a data set from the previously calculated values of the g parameter for contact elements that model the rolling elements. The value of the g parameter is a function of the current position of the table in the present system and depends on the geometric errors, which were adopted in the calculations. The computational simulation used an a priori function describing guide rail geometric errors according to the following equation: g=

2

gH =

5. Simulation of joint kinematic errors of a machine tool table

(2)

and for the g < 0 that is for the gap (shortening)



x ·+ 1000

 x 2  1000

·

2

where  = 0.05 mm, x in mm; the current position of the table. Taking into account the geometry of the rolling ball in contact with the rail and carriage, the aforementioned function was used in the later part of this paper in models for geometric errors of the guide rail, on vertical and horizontal planes. In this approach, such a system is subjected to kinematic excitations caused by geometric errors in the guide rail. We analyzed three variants of

2

gH =

c 2 + (b − i ) − d

 gV =

2

b2 + (c − i ) − d

(3)

where the H index refers to the geometric error defined in the horizontal plane, the V index refers to the geometric error defined in the vertical plane, d – diameter of the rolling element, and i – geometric deviation, b and c according to Fig. 5. The above described method is proposed as a solution that allows the simulative study of the effects of geometric errors occurrence in the linear guide joints.

(4)

Fig. 5. Geometric dependences in cross-section of carriages – guide assembly.

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Fig. 6. Table – guideway configuration (bottom side); (a) detailed model, (b) simplified model, and (c) model FEM.

calculations, which differ in the location of the error described by Eq. (4): • Variant 1 – one guide rail has a geometric error in the vertical plane and a second rail does not have any error. • Variant 2 – one guide rail has a geometric error in the horizontal plane and the second rail does not have any error.

• Variant 3 – one guide rail has a geometric error in the vertical plane and a second guide rail has a geometric error in the horizontal plane. The aforementioned variants were meant to create an image of the effect of geometric errors in the guide rails on joint kinematic errors. The results of a computational session for translational and angular errors for the table modeled using the aforementioned

Fig. 7. Characteristics of joint kinematic errors variant 1 (FEM calculation).

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Fig. 8. Characteristics of joint kinematic errors variant 2 (FEM calculation).

assumptions are shown in Figs. 7–9. The initial value (zero) table position against the guideways was adopted in the central part of the considered system. The current position of the table is on the X-axis; the value of the joint kinematic error is on the Y-axis. The values of all joint kinematic errors have been determined in the same coordinate system in which the FEM model has been defined. In the drawings, the icon schematically shows the location of the geometric error of the guide rail. Each of the characteristics of the joint kinematic error was made for five control points (P1–P5) on the surface of the table – in the four corners and the fifth point in the middle of the table. The results of calculations can be commented as follows. Since the guide rail geometric errors were a priori assumed, the obtained calculation results should be considered in qualitative terms. It should be remembered that comparisons are relevant to the characteristics of the computational gains obtained rather than absolute values. In this approach modeling applies to the linear guide connection with arbitrary assumed geometric errors. Therefore the results of calculations should not be directly compared with the results of experimental studies presented in Section 6 of further part of this paper. This approach is justified because the conclusions made on the results of computer simulations are to assist and to mark out the direction of experimental research rather than to describe the behavior of a particular machine tool. The system was considered as fixed on an ideally rigid surface, and therefore the system is probably stiffer in relation to the

real system. Hypothetically, this means that the occurrence of geometric errors in a real system, with similar values to those that were included in the calculations, would cause smaller increases in individual joint kinematic errors. Different characteristics of translational errors (positioning, vertical and horizontal straightness error) for options 1–3 in various control points on the surface of the table, indicate that the table as a solid undergoes rotation; while differing characteristics of the angular errors (pitch, yaw and roll error) indicate that the table is subject to a complex strain state, which in turn indicates a significant deviation in the behavior of the considered system from the assumed ideal stiffness. If this request is confirmed experimentally, it will mean that the deformation occurring in the system makes it impossible to define the spatial movement of solids in the form which can be used to model volumetric error [25–29]. In such case off-line geometric errors compensation should not be used. Analyzing the graphs of translational errors (three variants/option: Figs. 7–9), it can be stated that the directions of displacement of the various control points on the table are consistent with the nature of the defined geometric error – and seemingly do not pose a problem in interpretation. This state should be considered valid as the table as a solid may be experiencing rotation. In the analysis of translational errors it can be noted that if the guideway is subject to a dominant geometric error in the vertical plane (variant 1 in Fig. 7) the vertical straightness error is dominant (the greatest increases). Similarly, for a dominant error in the horizontal

P. Majda / Precision Engineering 36 (2012) 369–378

375

Fig. 9. Characteristics of joint kinematic errors Variant 3 (FEM calculation).

plane (variant 2 in Fig. 8), horizontal straightness error is dominant (the largest increases). If guide rails contain geometric errors on a similar level in the horizontal and vertical planes (variant 3 in Fig. 9), the dominant translation error is vertical straightness error, due to the greater flexibility in the vertical direction than in the horizontal direction (in the plane of the table). Positioning error plays the smallest part among the concerned translation errors. In real systems, its value is primarily determined by the static and dynamic properties of feed drives and the characteristics of the used displacement measurement systems. However, on the basis of the results of calculations, it can be concluded that classical measurement of positioning accuracy [24] may also include a term which is the effect of guideway geometric errors. Analyzing the graphs of angular errors (three variants: Figs. 7–9), it can be stated that the table is subject to a complex strain state. With regard to the models for volumetric errors, it is desirable for such a system to resemble the behavior of rigid solids. So it is desirable that the characteristics of angular kinematic errors, irrespective of the guideway geometric errors and irrespective of the position of a control point, are qualitatively and quantitatively similar. That occurs only for yaw error characteristics (Figs. 7–9) resulting from the geometry, i.e. in the plane normal to the surface of the table the values of second moment of area are much lower than in a plane parallel to the surface of the table. Thus we observe a greater flexibility on table bending in planes normal to the table surface, and much higher qualitative differences

in the characteristics of pitch and roll error, in comparison to yaw error. The results of calculation for pitch and roll error for variant 2 in Fig. 8 are very interesting, i.e. a variant where the dominant error is the geometric guide rail in the horizontal plane. We are seeing not only the quantitative changes of these characteristics, but also important qualitative changes. Yaw or roll errors can undergo an increase or decrease in value depending on the current position and depending on the location of the checkpoint in question on the surface of the table. This situation has very important implications for modeling volumetric error where it is desirable to have the characteristics of angular kinematic error clearly defined regardless of the location of the checkpoint on the surface of the table. If the simulation results are to be considered credible, the following question arises: is there a possibility of using the characteristics of angular kinematic errors in the correction of volumetric errors in machine tool? The answer is no. It is because in operating conditions, the trajectory of movement (with an angular kinematic error) associated with the workpiece or chuck, will depend on the location on the machine table. In this case, the use of these characteristics in order to increase the accuracy of an off-line machine is problematic because prior to fixing the workpiece on the table, angular kinematic errors are unknown. The practical use of the characteristics of such errors to compensate for machine tool geometric errors would require the measurement of their value for a particular position of a workpiece on the table.

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Fig. 10. Experimental characteristics of pitch error in the four corners of the table: (a) measurement with electronic levels, and (b) measurement taken with the laser interferometer.

It is characteristic that, regardless of the direction (vertical or horizontal) of the guideway geometric errors in the examined system, a roll error is always dominating (the largest increases) – Figs. 7–9. An interesting case can be observed for variant 2 and the roll error. Increases in this error, determined at the control point in the central part of the table, are close to zero. However, in the corners we can see a significant increase in the error. Assuming that such a situation could occur in the real object, then because in direct measurements it is usually measured in the central part of the table, the roll error would be underestimated. 6. Experimental research of angular kinematic errors

and in control points on the surface of the table, i.e. in its four corners. During measurement, the electronic level was placed at the control points and the reference level was placed on the spindle head Figs. 10–12 present the results of experimental measurements of angular kinematic errors in the four corners of the table. In the plots are photos showing the location of the measuring instruments during the measurement. Each characteristic is the averaging of five bidirectional measurements. Bidirectional repeatability for the characteristics, measured with the laser interferometer, did not exceed 2 ␮m/m, and for the levels 4 ␮m/m. The experimental measurements were carried out to verify the conclusions that were formulated on the basis of calculations of joint kinematic errors presented in Section 5, namely:

The subject of the study was the characteristics of angular kinematic errors of a milling machine table with a serial kinematic structure – XYOZ. Machining dimensions are 700/460/500 mm. In the sliding joints Mannesmann-Rexroth roller guideways (size 25) were used. Measurements were performed using a laser interferometer and electronic levels. Measurements were made on the table against the spindle head. The elements of the laser interferometer optics used in the measurement of angular misalignment were mounted in the spindle

• Is the tested table flexible enough to observe significant qualitative differences in the characteristics of angular kinematic errors for various control points? • Is there qualitative and quantitative conformity among the measured characteristics of angular errors, regardless of the location on the machine table? • Does yaw error confirm the nature of movement of the table as a rigid body? • Does roll error have a greater range in relation to pitch and yaw error?

Fig. 11. Experimental characteristics of yaw error in the four corners of the table – measurement with a laser interferometer.

Fig. 12. Experimental characteristics of roll error in the four corners of the table – measurement with electronic levels.

P. Majda / Precision Engineering 36 (2012) 369–378

• With regard to flexible structures, is there a possibility of using the results of direct measurement angular kinematic errors to compensate for the volumetric error? Yaw error cannot be measured with electronic levels, so Fig. 11 shows only the result of the measurement taken with a laser interferometer. A similar situation occurs with roll error – Fig. 12, which is easily measured with levels, and its measurement by an interferometer is cumbersome without a special set of optics. The obtained results of measurements of angular kinematic errors in the four corners of the table confirm conclusions based on the results of analytical calculations, i.e.: • The tested table showed higher flexibility than the carriages on which it was mounted. It is shown by the different quantitative and qualitative characteristics of angular kinematic errors that result from strains to kinematic excitation resulting from guide geometric errors. This applies especially to roll error (Fig. 12) which, depending on the table, not only changes value but also even the direction of increase. • Regardless of the position of the laser interferometer on the table, measurements confirmed the qualitative and quantitative similarity of yaw error characteristics – Fig. 11. Therefore it confirms the conclusions resulting from computer simulation, i.e. the insensitivity of the examined system to strains in the plane of the table. • Pitch and yaw errors reach 20 ␮m/m, while the roll error is in the range from 8 to 32 ␮m/m. Thus it confirms that the roll error is predominant among angular kinematic errors for a table that can be subject to strain. • The use of the presented characteristics in compensation for volumetric error is questionable. Out of the three measured errors, only yaw error does not raise objections, because it confirms the assumption that the table is subject to rigid body motion. Characteristics of pitch error, due to the point for which it is considered, show a significant dispersion, but it qualitatively confirms the direction of the movement of a solid, so it seems that the use of this characteristic in compensation is also justified. However, in this case, the use of roll error is unacceptable in the model for volumetric error compensation because its value and even the direction depend on the location in which it is measured on the machine table. This demonstrates the systematic nature of this error only for a specific configuration in which a measurement was made. One should be aware that in operating conditions the measuring apparatus will be replaced by a workpiece, the unwanted movement of which, due to the machine tool geometric errors, will depend on the location on the table, i.e. before fixing the roll error characteristics would be unknown, and thus it cannot be practically used to compensate for off-line volumetric error in machine tool. 7. Conclusion This paper proposes a model for rolling guideways with geometric errors, and considers aspects of the practical use of the characteristics of joint kinematic errors in models for volumetric error in a medium-sized machine tool. It has been shown analytically and confirmed experimentally, that the strains of the table due to guideway geometric errors should be taken into account during mapping of geometric errors. Development in methodology of volumetric error determination in the case of elastic structures is still an open question. It should be emphasized that the conclusions of this work are of fundamental importance for the methodology of measuring and determining volumetric error in machine tool with movable tables.

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Most of the known measuring methods require that the measuring instruments should be set on the machine table. It has been shown that such a situation may be accompanied by ambiguity in the determination of certain angular error characteristics (especially roll error). Therefore, it is problematic to adopt a representative configuration for the measurement of these characteristics, and thus it is also debatable to use them later to compensate for offline volumetric error in machine tools. Solutions to this problem must be sought in the development of methods of measuring volumetric error on-line or developing new design solutions that allows to control the movement of machine tool parts in an easy way (only six degrees of freedom). Acknowledgment The work was financed from the Resources for National Science Centre as a research project no. N N504 670440. References [1] Brecher C, Utsch P, Klar R, Wenzel C. Compact design for high precision machine tools. International Journal of Machine Tools & Manufacture 2010;50:328–34. [2] Woody BA, Smith KS, Hocken RJ, Miller JA. A technique for enhancing machine tool accuracy by transferring the metrology reference from the machine tool to the workpiece. Journal of Manufacturing Science and Engineering 2007;129:636–43. [3] http://mtpselector.renishaw.net/en/vertical-machining-centres – 8458. [4] Schwenke H, Knapp W, Haitjema H, Weckenmann A, Schmitt R, Delbressine F. Geometric error measurement and compensation of machines – an update. CIRP Annals – Manufacturing Technology 2008;57:660–75. [5] Yang SH, Kim KH, Park YK, Lee SG. Error analysis and compensation for the volumetric errors of a vertical machining centre using a hemispherical helix ball bar test. International Journal of Advanced Manufacturing Technology 2004;23:495–500. [6] Bringmann B, Knapp W. Machine tool calibration, geometric test uncertainty depends on machine tool performance. Precision Engineering 2009;33:524–9. [7] Choi JP, Minb BK, Lee SJ. Reduction of machining errors of a three-axis machine tool by on-machine measurement and error compensation system. Journal of Materials Processing Technology 2004:155–6. [8] Schwenke H, Franke M, Hannaford J. Error mapping of CMMs and machine tools by a single tracking interferometer. CIRP Annals – Manufacturing Technology 2005;54:475–8. [9] Schwenke H, Schmitt R, Jatzkowski P, Warmanna C. On-the-fly calibration of linear and rotary axes of machine tools and CMMs using a tracking interferometer. CIRP Annals – Manufacturing Technology 2009;58:477–80. [10] Ahn KG, Cho DW. An analysis of the volumetric error uncertainty of a three axis machine tool by beta distribution. International Journal of Machine Tools and Manufacture 2000;40:2235–48. [11] Okafor AC, Ertekin YM. Vertical machining center accuracy characterization using laser interferometer. Part 1. Linear positional errors. Journal of Materials Processing Technology 2000;105:394–406. [12] Raksiri C, Parnichkun M. Geometric and force errors compensation in a 3-axis CNC milling machine. International Journal of Machine Tools and Manufacture 2004;44:1283–91. [13] Ekinci TO, Mayer JRR. Relationships between straightness and angular kinematic errors in machines. International Journal of Machine Tools and Manufacture 2007;47:1997–2004. [14] Ferreira PM, Liu CR. Contribution to the analysis and compensation of the geometric error of a machining center. CIRP Annals 1986;35(1):259–62. [15] Srivastava AK, Valdhuis SC, Elbestawit MA. Modelling geometric and thermal errors in a five-axis CNC machine toll. International Journal of Machine Tools and Manufacture 1995;35(9):1321–37. [16] Jung JH, Choi JP, Lee SJ. Machining accuracy enhancement by compensating for volumetric errors of a machine tool and on-machine measurement. Journal of Materials Processing Technology 2006;174:56–66. [17] Pahk HJ, Kim JS, Moon J. A new technique for volumetric error assessment of CNC machine tools incorporating ball bar measurement and 3D volumetric error modelint. International Journal of Machine Tools and Manufacture 1997;37(11):1583–96. [18] Zienkiewicz OC. The Finite Element Method. New York: McGraw-Hill; 1977. [19] Dhupia JS, Ulsoy AG, Katz R, Powalka B. Experimental identification of the nonlinear parameters of an industrial translational guide for machine performance evaluation. Journal of Vibration and Control 2008;14(5):645–68. [20] Hung JP, Lai YL, ChY. Lin, Lo TL. Modeling the machining stability of a vertical milling machine under the influence of the preloaded linear guide. International Journal of Machine Tools & Manufacture 2011;51:731–9. [21] Wu JSS, Chang JCh, Hung JP. The effect of contact interface on dynamic characteristics of composite structures. Mathematics and Computers in Simulation 2007;74:454–67.

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